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Filtration shrinkage, the structure of deflators, and failure of market completeness

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  • Constantinos Kardaras
  • Johannes Ruf

Abstract

We analyse the structure of local martingale deflators projected on smaller filtrations. In a general continuous-path setting, we show that the local martingale part in the multiplicative Doob-Meyer decomposition of projected local martingale deflators are themselves local martingale deflators in the smaller information market. Via use of a Bayesian filtering approach, we demonstrate the exact mechanism of how updates on the possible class of models under less information result in the strict supermartingale property of projections of such deflators. Finally, we demonstrate that these projections are unable to span all possible local martingale deflators in the smaller information market, by investigating a situation where market completeness is not retained under filtration shrinkage.

Suggested Citation

  • Constantinos Kardaras & Johannes Ruf, 2019. "Filtration shrinkage, the structure of deflators, and failure of market completeness," Papers 1912.04652, arXiv.org, revised Aug 2020.
    • Handle: RePEc:arx:papers:1912.04652
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    References listed on IDEAS

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    1. Acciaio, Beatrice & Fontana, Claudio & Kardaras, Constantinos, 2016. "Arbitrage of the first kind and filtration enlargements in semimartingale financial models," Stochastic Processes and their Applications, Elsevier, vol. 126(6), pages 1761-1784.
    2. Acciaio, Beatrice & Fontana, Claudio & Kardaras, Constantinos, 2016. "Arbitrage of the first kind and filtration enlargements in semimartingale financial models," LSE Research Online Documents on Economics 65150, London School of Economics and Political Science, LSE Library.
    3. Delia Coculescu & Monique Jeanblanc & Ashkan Nikeghbali, 2012. "Default times, no-arbitrage conditions and changes of probability measures," Finance and Stochastics, Springer, vol. 16(3), pages 513-535, July.
    4. Jeanblanc, Monique & Song, Shiqi, 2015. "Martingale representation property in progressively enlarged filtrations," Stochastic Processes and their Applications, Elsevier, vol. 125(11), pages 4242-4271.
    5. Claudio Fontana & Monique Jeanblanc & Shiqi Song, 2014. "On arbitrages arising with honest times," Finance and Stochastics, Springer, vol. 18(3), pages 515-543, July.
    6. Fontana, Claudio, 2018. "The strong predictable representation property in initially enlarged filtrations under the density hypothesis," Stochastic Processes and their Applications, Elsevier, vol. 128(3), pages 1007-1033.
    7. Prokaj, Vilmos, 2009. "Unfolding the Skorohod reflection of a semimartingale," Statistics & Probability Letters, Elsevier, vol. 79(4), pages 534-536, February.
    8. Anna Aksamit & Tahir Choulli & Jun Deng & Monique Jeanblanc, 2017. "No-arbitrage up to random horizon for quasi-left-continuous models," Finance and Stochastics, Springer, vol. 21(4), pages 1103-1139, October.
    9. Ioannis Karatzas & Constantinos Kardaras, 2007. "The numéraire portfolio in semimartingale financial models," Finance and Stochastics, Springer, vol. 11(4), pages 447-493, October.
    10. Song, Shiqi, 2016. "Drift operator in a viable expansion of information flow," Stochastic Processes and their Applications, Elsevier, vol. 126(8), pages 2297-2322.
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    Cited by:

    1. David Criens & Mikhail Urusov, 2023. "Criteria for the absence of arbitrage in general diffusion markets," Papers 2306.11470, arXiv.org, revised Sep 2024.
    2. Francesca Biagini & Andrea Mazzon & Ari-Pekka Perkkiö, 2023. "Optional projection under equivalent local martingale measures," Finance and Stochastics, Springer, vol. 27(2), pages 435-465, April.
    3. Černý, Aleš & Ruf, Johannes, 2023. "Simplified calculus for semimartingales: Multiplicative compensators and changes of measure," Stochastic Processes and their Applications, Elsevier, vol. 161(C), pages 572-602.
    4. Çetin, Umut & Larsen, Kasper, 2023. "Uniqueness in cauchy problems for diffusive real-valued strict local martingales," LSE Research Online Documents on Economics 118743, London School of Economics and Political Science, LSE Library.
    5. Umut Cetin & Kasper Larsen, 2020. "Uniqueness in Cauchy problems for diffusive real-valued strict local martingales," Papers 2007.15041, arXiv.org, revised May 2022.

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